My question: Is it possible to make a frame of an animation (let’s say frame 30 of the default 0-30 timeline) as the new frame 0 of a fresh “clean” animation? And, is it possible to save a specific frame as a scene, without the other frames?

If you want to start from a specific frame, go to that frame, select everything, and open the Parameters pane option menu (click the button in the top corner, or in DS 4 right-click the tab) and from the menu clear the Animation data from the items - they will then be in the same state in every frame, with no keyframes set, so you can go to frame 0, set a keyframe with the Timeline button and go on from there. I think this would be the only way to save a scene at a particular frame, too, though all presets for animatable properties will allow you to set a frame range to be included.

You can use “Puppeteer” to capture any poses, (keyframes) and you can save those as a collection of poses.
If you move the timeline to any position, then click one of those saved pose dot’s, then it will apply that pose to the figure.

You can also select any point in the timeline, and go to File / Save as / Pose preset.